For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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0answers
48 views

Conditional expectation of discrete uniform random variables with one fixed

Came across a problem that I worked on sometime ago having the following structure: Given an opaque container (or locomotive with so many passenger cars, etc) that has--with equal probability--1 to N ...
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0answers
45 views

understanding uniform distribution on multiset

If I have a set of words that form a text, say text $a$, and a set of texts, I then calculate the similarity between text $a$ and each text in the set. Then, I get a multiset. For example: $$s = ...
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1answer
190 views

what is the probabilty that sum of two random numbers between A and B is less than third number C [closed]

What is the probabilty that sum of two random numbers uniformly distributed in $[A,B]$ is less than a fixed $C$? I have tried answering this question using graph method to find the area under the ...
3
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0answers
37 views

How to sample from a convex hull?

Let us consider a couple of points $x^{(i)}\in \mathbb{R}^m$ where $i=1,\dots,n$. Convex hull is defined as $$ C = \left\{\sum_{i=1}^{m} \alpha_i x^{(i)} \mathrel{\Bigg|} (\forall i: \alpha_i\ge ...
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0answers
53 views

Converting normally distributed numbers to uniform distribution

I have a Perlin noise algorithm I've written my self. It seems to produce gausian numbers at the range of -1.5 and 1.5 but I'll convert them to the range of -1 and 1. I' currently working on a project ...
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0answers
32 views

Product of n uniformly distributed RVs

Let $X_j \sim U(a,b)$. What is the PDF of $\prod_{j=1}^n X_j$? I have seen some with $X_j \sim U(0,1)$ but I was wondering what the general form of the solution is for any $a$ and $b$.
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5answers
75 views

PDF of function of uniform random variable [closed]

Why PDF of $g(X)=X^3$ is not uniformly distributed, when X is uniform random variable between $(0,1)$? As for every value of X there is unique value of $g(X)$, hence the probability density of $g(X)$ ...
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3answers
54 views

If $X$~$U(0,1)$, and $Y=2x-4$. What is the density function of Y?

If $X$ is uniformly distributed $\mathcal{U}(0,1)$ , then what is the distribute density function of $Y$? I thought that if $$fx(x) = 1/(1-0), \; \mbox{for} \; 0<x<1$$ then ...
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0answers
29 views

Understanding compound distribution and plotting the mixed distribution graph

If $N$ takes the values $0, 1$ and $2$ with probabilities $½, ¼ $ and $¼ $ respectively, and the $X_i$ ’s have a $U(0,10)$ distribution, draw a sketch of the frequency distribution of $S$. $N$ is the ...
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1answer
21 views

Finding a uniform distribution on the output of a multivariable function

Suppose we have a non-invertible continuous function that maps from some continuous interval ${I}^n$ to $\mathbb{R}$ with $n \ge 1$. To take an example, let $f(a,b,c) = a \cdot e^{-bc} - b \cdot ...
3
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2answers
116 views

A single, good test for a random number generator?

I'm chasing a bug in the RNG for a well-known programming language under certain pathological inputs. There is an obvious pattern in this pathological case, apparent with very small n (~ 10000), and ...
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0answers
35 views

How to reduce the standard deviation from $P$ where $P$ involves a random integer uniformly distributed in $\left[ 0,100\right]$

I have a probability $P$ derived from: - A random integer $A$ uniformly distributed on its range such that $A\in\left[0, 100\right]$ - An integer $K$ such that $K\in\Bbb N$ - A number $X$ such that ...
1
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1answer
26 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
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2answers
213 views

Why birthday distribution is not uniform. [closed]

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
1
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3answers
65 views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
3
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1answer
53 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...
2
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1answer
55 views

Non-Linear System. Find the conditional expectation.

I've had my test for this course and I think I failed it again. The hardest part for me is findig the correct distributions. This is a test exercise I couldn't figure out or at least, I probably ...
2
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0answers
37 views

Is there a connection between uniform law of large number and Ibragimov's conjecture?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following: Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, ...
2
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0answers
40 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
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2answers
64 views

Uniform distribution on $\{1,\dots,7\}$ from rolling a die [duplicate]

This was a job interview question someone asked about on Reddit. (Link) How can you generate a uniform distribution on $\{1,\dots,7\}$ using a fair die? Presumably, you are to do this by combining ...
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1answer
37 views

Uniform distribution on sphere

Let $U = (u_1, u_2, u_3)$ is random vector uniformly distributed on unit sphere $S^{2} \subset \mathbb{R}^3$. Are $u_1, u_2, u_3$ mutually independent ? I guess not, but I have no idea to prove it.
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1answer
63 views

Distribution of the difference of order statistics

Let $X_1$ and $X_2$ be a random sample of size 2 from a uniform distribution over the interval $[0,1]$, let $Y_1$ and $Y_2$ be the corresponding order statistics. Find the conditional density ...
2
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2answers
72 views

The probability density of $X^2$?

Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? ...
0
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1answer
39 views

Calculate $P(|X-4| > 1.5)$

If $X \sim U(2, 8)$ Would it be $$P(X > 1.5 + 4) + P(X <-1.5 +4)$$
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2answers
56 views

Independence, conditioning, and correlations

Suppose $X$ and $Y$ are independent random variables uniformly distributed on $[0,1]$. Suppose we consider a conditional distribution of $X$ and $Y$ on some event $C$. Is it possible that these ...
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0answers
40 views

Exlain the significance of the uniform random variable for the simulation of random variables

I can think of the "Universality of the Uniform": Given an Unif(0,1) r.v., we can construct an r.v. with any cts distribution we want. Conversely, given an r.v. with an arbitrary cts ...
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1answer
23 views

unbiased estimator of $m^2$ of uniform distribution over $(0,m)$ [closed]

I have a sample of size $1$ from a distribution that has $U\left(4,4+m\right)$ as probability density function. Is there any unbiased estimator for $m^{2}$?
2
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0answers
49 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
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1answer
100 views

Limit moment generating function

For n a natural number let $X_{n}$ have discrete uniform distribution on interval {1,2...,n} and $Y_{n} =\frac{1}{n} X_{n}$. I need to show that for all t(real number) the $\lim_{n \to \infty} ...
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1answer
54 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
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1answer
121 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
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0answers
50 views

hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
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2answers
405 views

Average distance from center of circle to evenly-distributed points within it

With some number of points that are evenly/uniformly (assuming those mean the same thing) distributed within a circle of radius 1, what is the average distance from the center of the circle to a ...
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0answers
47 views

Expectation of $v=\inf \{n\geq 2\,;\, X_n > X_1 \}$ when $(X_n)$ is i.i.d. uniform on (0,1)

Let $W$ be the occurence meaning the following ordering : $X_1...X_k$ where $X_k$ is greatest.. $X_k$ is greatest, and next in order is $X_1$, and the order of the others is not important. Because of ...
2
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2answers
55 views

Maximum likelihood estimator on uniform distribution

I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me mad. Here it goes. Consider the uniform distribution on ...
2
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1answer
30 views

Maximum likelihood on uniform distribution

In a exercise i'm doing it is asked to find the maximum likelihood estimator of a random sample $X_{1}, ... , X_{n}$ of a population with distribution $X\sim U(- \theta , \theta) $. I've found that ...
2
votes
1answer
213 views

is there a concept of asymptotically independent random variables variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...
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1answer
68 views

Bivariate probability distribution(s) over unit square, uniform marginals, midpoint is saddlepoint

Construct a bivariate probability distribution--or family of such distributions--over the unit square (corners $(0,0), (0,1), (1,1), (1,0)$) with uniform marginals and having a saddlepoint at ...
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1answer
52 views

Distribution of function of a Random Variable

If $X$ is uniform on $(0,1)$, how would I go about finding the CDF of $Y=(X-X^2)^2$ ?Thanks.
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1answer
130 views

Uncertain about which probability method to use for the problem

Suppose I want to catch a bus (which runs every 10 minutes on average). What is the probability that: 1). You will wait for at least fifteen minutes before the bus arrives, and then, 2). 3 buses ...
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0answers
34 views

Conditional expectations with identical marginals and positively dependent but unknown joint distribution

Let $A$ and $B$ be random variables, each with marginal distribution $% U\left( 0,1\right) $, but unknown joint distribution $H\left( a,b\right) $. Suppose $A$ and $B$ are each stochastically (weakly) ...
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0answers
9 views

Sampling specific unit vectors

Given a unit vector $A\in \Bbb{R}^N$ and an angle $\theta$, the unit vector $P$ needs to satisfy $\left<A,P\right>=\cos\theta$. How to sample $P$ uniformly? For example: If $A = ...
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1answer
26 views

Optimal Value & Uniform Distribution

In a simple setting, $w$ is uniformly distributed on $[0,1]$, R is a function of $wd$. I want to find optimal d in this expression, $aR-(d^2-1)/2$. When I try to find out optimal $d$ than it is $0$. ...
2
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0answers
37 views

Inconsistent answers with conditional expectations

Let $X$ and $Y$ be two independent random variables distributed uniformly on $[0,1]$. I want to compute: $$E[X+Y\mid\max\{X,Y\}≤(1/2)]$$ My first approach was the following. Let $X=\max\{X,Y\}$. ...
2
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2answers
229 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
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2answers
68 views

Calculating the expected value of a random variable that's a function of a random variable

I am working on the following problem: I'm having a hard time putting all of this information together: The cost of the maintenance is $Z = X + Y$, where $X$ is the cost of the first machine and ...
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0answers
26 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
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1answer
41 views

Demonstrate uniform continuous distribution using tangible items?

What is the best way to explain "equally likely" in continuous uniform distribution to an audience using tangible or everyday items?
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2answers
134 views

Calculating if a point is within the overlap of two circles

Two circles of equal radius (R) intersect as shown below. Assuming more points are uniformly distributed in an area with dimensions D*D, where D = 4*R. What is the probability that a point will be ...
2
votes
1answer
43 views

Stoppage time for sequence of uniform random numbers with a recursively shrinking domain

Define $x_n = U(x_{n-1})$ where $U(x)\in\lbrace 0,1,\ldots,x\rbrace$ is a uniformly distributed random integer. Given $x_0$ as some large positive integer, what is the expected value of $n$ for which ...